| L(s) = 1 | − 0.806·3-s + 5-s − 3.35·7-s − 2.35·9-s − 0.962·11-s + 6.15·13-s − 0.806·15-s − 6.31·17-s + 19-s + 2.70·21-s + 4.96·23-s + 25-s + 4.31·27-s − 3.61·29-s + 5.92·31-s + 0.775·33-s − 3.35·35-s + 10.1·37-s − 4.96·39-s + 6.31·41-s + 4.12·43-s − 2.35·45-s − 3.35·47-s + 4.22·49-s + 5.08·51-s + 1.84·53-s − 0.962·55-s + ⋯ |
| L(s) = 1 | − 0.465·3-s + 0.447·5-s − 1.26·7-s − 0.783·9-s − 0.290·11-s + 1.70·13-s − 0.208·15-s − 1.53·17-s + 0.229·19-s + 0.589·21-s + 1.03·23-s + 0.200·25-s + 0.829·27-s − 0.670·29-s + 1.06·31-s + 0.135·33-s − 0.566·35-s + 1.66·37-s − 0.794·39-s + 0.985·41-s + 0.629·43-s − 0.350·45-s − 0.488·47-s + 0.603·49-s + 0.712·51-s + 0.253·53-s − 0.129·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.177794894\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.177794894\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 0.806T + 3T^{2} \) |
| 7 | \( 1 + 3.35T + 7T^{2} \) |
| 11 | \( 1 + 0.962T + 11T^{2} \) |
| 13 | \( 1 - 6.15T + 13T^{2} \) |
| 17 | \( 1 + 6.31T + 17T^{2} \) |
| 23 | \( 1 - 4.96T + 23T^{2} \) |
| 29 | \( 1 + 3.61T + 29T^{2} \) |
| 31 | \( 1 - 5.92T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 - 6.31T + 41T^{2} \) |
| 43 | \( 1 - 4.12T + 43T^{2} \) |
| 47 | \( 1 + 3.35T + 47T^{2} \) |
| 53 | \( 1 - 1.84T + 53T^{2} \) |
| 59 | \( 1 - 6.38T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 - 6.73T + 67T^{2} \) |
| 71 | \( 1 - 0.775T + 71T^{2} \) |
| 73 | \( 1 - 0.387T + 73T^{2} \) |
| 79 | \( 1 - 0.836T + 79T^{2} \) |
| 83 | \( 1 - 7.03T + 83T^{2} \) |
| 89 | \( 1 - 7.08T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.256535066609790094058231845723, −8.954524838498873351040686739130, −7.932160086340054580401837564389, −6.63144874542798838372369477283, −6.28561856886506231686131666840, −5.59562810747179178073161718668, −4.44147398563022317296780608891, −3.33193957116151940061608630837, −2.48525198776378497071493119436, −0.77367532246184838943517958557,
0.77367532246184838943517958557, 2.48525198776378497071493119436, 3.33193957116151940061608630837, 4.44147398563022317296780608891, 5.59562810747179178073161718668, 6.28561856886506231686131666840, 6.63144874542798838372369477283, 7.932160086340054580401837564389, 8.954524838498873351040686739130, 9.256535066609790094058231845723
